Independent Variate Stream Generator
Produces continuous streams of independent identically distributed random variables from Bernoulli, Uniform, and Cauchy parents.
Statistics & Probability / Law of Large Numbers
FREE TO EXPLORELABORATORY 06 · STATISTICS & PROBABILITY
Weak and Strong Law of Large Numbers (Bernoulli & Khinchin)
Investigate the Weak Law of Large Numbers (WLLN) and Strong Law of Large Numbers (SLLN), contrast convergence in probability with almost sure convergence, and verify the precision scaling of sample means. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
RECIPROCAL VARIANCE 1 / VAR(X̄)—
SAMPLING TELEMETRY
Ready to sample with Sample Size N = 10.0.
Adjust Sample Size N, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
The LLN guarantees asymptotic convergence as N → ∞ but provides no error bounds for any single finite sample size N. The empirical regression confirms the theoretical precision scaling 1/Var(X̄_N) = 1.000 · N with R² = 1.000000 and 0.0000% error.
Statistical tools & sampling modules.
Running Sample Mean Accumulator
Computes sequential partial sums S_n = ∑ X_i and cumulative averages X̄_n = S_n / n along individual sample trajectories.
Sample Path Trajectory Tracker
Monitors multiple parallel realization paths to visualize almost sure sample-path convergence without barrier escape.
Convergence in Probability Density Meter
Computes cross-sectional empirical variances Var(X̄_N) across 1,000 independent Monte Carlo trials at fixed N.
Cauchy Counterexample Comparator
Tests the LLN on heavy-tailed Cauchy variates to observe infinite-variance trajectory divergence.
Linear Precision Regression Unit
Performs linear regression of inverse sample mean variance 1/Var(X̄_N) versus N to verify unit slope m = 1.000.
FOLLOW THE METHOD
A reason behind every step.
Investigate the Weak Law of Large Numbers (WLLN) and Strong Law of Large Numbers (SLLN), contrast convergence in probability with almost sure convergence, and verify the precision scaling of sample means. Standardized parameter settings: 10.0, 20.0, 30.0, 40.0, 50.0, 60.0 trials.
01
Initialize Independent Random Parent Process
Configure an i.i.d. random generator for a zero-mean distribution with unit variance σ² = 1.0 (such as standard normal or centered Bernoulli p = 0.5).
02
Simulate Parallel Sample Paths
Execute 1,000 parallel Monte Carlo sample path trajectories up to N = 100 steps, recording sequential running means X̄_n at each step n.
03
Measure Cross-Sectional Variance at N = 10
Set sample size N = 10. Extract the cross-sectional sample means across all 1,000 trajectories. Compute the sample mean variance Var(X̄_10) = 0.1000 and precision 1/Var(X̄_10) = 10.00.
04
Log Observation at Baseline N = 10
Record the baseline coordinate (x = 10.0, y = 10.0) in the laboratory notebook, noting tight concentration around the theoretical mean μ = 0.
05
Step Sample Size Across N = 10 to 60
Increment sample size across standardized increments N ∈ {10, 20, 30, 40, 50, 60}, observing the linear increase in estimation precision and suppression of trajectory dispersion.
06
Perform Linear Regression Analysis
Fit a linear regression model y = mx + b to the data pairs (x = N, y = 1/Var(X̄)). Confirm that the empirical slope m = 1.0000 exactly verifies the theoretical variance relation Var(X̄) = σ²/N.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Law of Large Numbers mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.0000 · x
| # | Sample Size N (trials) | Reciprocal Variance 1 / Var(X̄) (precision) | Sample Size N | Sample Mean Precision 1 / Var(X̄) | Remove |
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